Speaking to German public broadcaster ARD, Yildirim insisted that Turkish courts are independent but said: "I hope he will soon be set free."
According to a transcript of the interview to be aired in the evening, he added: "I believe there will be a development shortly."
The incarceration of the Die Welt daily correspondent has been the biggest stumbling block to rebuilding badly damaged ties between the governments of Chancellor Angela Merkel and President Recep Tayyip Erdogan.
Germany, home to a three-million-strong ethnic Turkish minority, has strongly criticised Erdogan's rights record, especially after a mass crackdown in response to a failed coup against him in July 2016.
Many German newspapers marked Yucel's first full year in custody without charge with frontpage stories Wednesday, when the #FreeDeniz campaign held a major solidarity event in Berlin.
Supporters of Yucel and other jailed reporters drove through central Berlin in a loudly honking motorcade of several dozen cars and bicycles, decorated with red heart-shaped balloons, joined by a bus of rights group Amnesty International.
'Strained relations'
Yucel, 44, appearing unbowed by his time behind bars, has written a book entitled "We're Not Here for Fun" which was to be launched at a star-studded reading in the German capital from 1900 GMT.
No formal charges have been laid, and no trial date has been set for Yucel, whom Turkish leaders have accused of "terrorist propaganda".
Yucel gave himself up to police on February 14, 2017 after initially taking refuge inside a German diplomatic compound in Istanbul when media reported an arrest warrant had been issued against him. He has been in custody ever since.
Die Welt reacted to Yildirim's stated hope that Yucel would soon be released from custody, with a journalist commenting that "here we really have something in common".
Merkel's spokesman Steffen Seibert reiterated Monday that the cases of Yucel and other Germans behind bars in Turkey for political reasons "put a strain on relations between Germany and Turkey".
"The chancellor has stated very clearly over the past year that we expect Turkey to release German citizens who are being detained for unjustifiable reasons," he said. "Deniz Yucel is the most prominent among them."
"We are doing everything we can, politically, diplomatically, to see an improvement in this case."
Relations between the NATO partners have recently started to recover from their deep crisis, in which Berlin last year advised German investors and holiday-makers to avoid the country, after Turkey released several German citizens.
German Foreign Minister Sigmar Gabriel said Wednesday that, after he had repeatedly discussed the Yucel case with his Turkish counterpart Mevlut Cavusoglu, he hoped for "a positive decision soon by the independent Turkish court".
Yildirim, ahead of his meeting with Merkel on Thursday, said in the ARD interview that the two countries "should turn over a new page, forget the past, look to the future and further build up our relationship".
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